Repeating Ground Track Orbit - poliastro/poliastro GitHub Wiki
This is a list of all the algorithms taken into consideration to achieve a repeating ground track orbit.
Possible solutions:
David Eagle (MATLAB Central)
Here we have four possible solutions to design a repeating ground track orbit:
- Time to repeat ground track (nodal period) using Kozai orbit propagation.
- Time to repeat ground track using numerical integration. This solution uses a Runge-Kutta-Fehlberg
- Required mean semi-major axis using Wagner's algorithm. This one, it's quite similar to what it's explained in Vallado's book, Fundamentals of Astrodynamics and Applications 4th edition, pages (869-873). The PR #984 implements the algorithm detailed in Vallado.
- Required osculating semi-major axis using numerical integration. It's like the one mentioned in
2.
, but with the variant that the time to each ascending node is also determined by a Runge-Kutta-Fehlberg algorithm.
Curtis
- Curtis has a different approach (Pages 227-229), computing
RA
yDec
relative to the rotating Earth after a time intervalDt
. It does not consider any perturbations.
Noreen 2003
- Noreen 2003 seems that the iterative algorithm taken from Vallado's doesn't seem to work with all the bodies. We can a find a slightly different solution to Mars.
Vtipil 2012
- Vtipil 2012 contains some mathematical background about the assumptions behind some other methods: Mortari "Flower constellations" (J2 only, any elliptic eccentricity), Aorpimai 2007 (J2, J4 and J2 ** 2, near-circular orbits) Vtipil 2010 and mentions a "simplified repeat groundtrack" (Collins 1977, J2).
All the examples to test the algorithm from PR #984 were obtained from:
- Handbook of Satellite Orbits From Kepler to GPS by Michel Capderou.
- Space Mission Analysis and Design, 3rd Edition, page 156. Specifically for the Earth case.
After some discussion, we agreed on implementing the Noreen algorithm for Sun-synchronous orbits to begin.
Assumptions:
- Noreen paper
- Circular orbits (no eccentricity)
- Sun-synchronous orbit (link between semimajor axis and inclination)
Notes:
- For Sun-synchronous orbits, if ecc = 0 we can fix a (and then i is given) or viceversa
- For RGT, Sun-synchronous orbits, ecc = 0 comes from the assumptions, a comes from the period requirement, and this fixes i
- Inputs:
k, ndays, norbits
(assumesecc = 0
) Q = Number of periods (norbits)
,P_Q = Time to complete 1 of the Q periods (nodal period?)
,P_sid = Mars sidereal rotation
,P_sol = Mars Solar rotation
=> Equation 2 givesP_Q
in terms of the (1) sidereal and (2) solar rotation of the body and (3) the number of orbits- There's a relationship between
P_K
anda
, and betweenn
anda
(third Kepler law) - Equation 10 from Noreen is similar to the second equation from Vallado pp. 870 (unnumbered): it relates the nodal period and the anomalistic period
P_Q = \frac{2 \pi}{n_t} = \frac{2 \pi}{n} \frac{n}{n_t} = &
P_K \frac{n}{n_t} = ...
but Noreen includes a factor cos(i)
that is not present in Vallado.
- Eqs. 6 to 8 are obtained with some assumptions with J2
Algorithm
- Inputs:
Q, k, planet rotations, (ecc = 0)
Q => P_Q
from Eq. 2 assuming retrograde orbit (90º <= inc <= 180º)P_Q => n_t
from Eq. 10- Eqs. 15 and 18 relate (i, a). In Eq. 18,
P_Omega = \frac{P_sol}{Q}
andP_K = 2 \pi \sqrt{\frac{a^3}{k}}
. But notice that Eq. 15 givescos i
directly, therefore it can be replaced inside Eq. 18 and this gives a relation betweena
and the input parameters of the problem (Q
and the rotation of the attractor). Therefore, we solve fora
. - We use
a
in Eq. 15 to calculatei
- Check that the orbit is retrograde (90º <= inc <= 180º), and if not, ???
- For the moment, raise an error